Soft and Collinear Behaviour of Graviton Scattering Amplitudes

Soft and Collinear Behaviour of Graviton Scattering Amplitudes • David Dunbar, Swansea University

Soft theorems • Part of General exploration of singularities of scattering amplitude as route to computation and comprehension • singularity as a leg (n) becomes soft Weinberg, 65 • • • soft factor is universal receives no loop corrections sub-leading terms are finite: for real momenta

Sub-leading terms are singularities in complex momenta : engineering a cubic singularity s

Soft Theorems Cachazo and Strominger White (subleading) Bern, Davies, Nohle

-Beyond the trees? =0 • • +other Soft theorem consequence (Ward identity) of BMS symmetry Bondi, van der Burg Metzner Sachs leading term protected

Gravity MHV amplitude Berends Giele, Kuijf, 87; Mason Skinner, 2009 dependance upon Hodge, 2011

MHV “Twistor-link”-representation Nguyen, Spradlin, Volovich, Wen, 2010 n=5 n=6 n=7 connected tree diagrams involving positive helicity legs only a b

Alternate Formulation From a Seed a b =

-soft lifting from three and four point tree

Alternate Formulation: 2 From Seeds

Soft-Terms from diagrams n-1 -point diagram t-dependance lies purely on green line

-diagram with soft leg attached to outside -summing contributions gives leading soft factor

diagrams with soft leg between two legs are pure quadratic

-diagrams with trivalent vertex for soft leg are pure linear divergent B A C -this matches {A B C A C B B A C }

N=4 One-loop, MHV n-point

-softlifting rational term? a b =

Collinear limit : nsatz satisfies leading soft behaviour but fails collinear li -need to add extra term -trivial when looked at the right way

N=4 One-loop, MHV n-point Rn is obtained by summing all link diagrams with a single loop a b =

-sub-leading soft gives “anomaly” • sub-leading soft can replace role of collinear limit in determining structure

Soft-Theorems for One-loop amplitudes • • • not many amplitudes available! N=8 : all available M(+++. . . ++++) N=6, 4 MHV pure gravity 4 pt+5 pt completely. . use what we have

Passarino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator

Finite Loop Amplitudes Bern Dixon Perelstein Rosowsky, 98

Single Minus, double poles

double Poles • • for real momenta amplitudes have single poles double poles arise when we use complex momenta + + a + b

-double poles not intrinsically a problem • but we need a formula for sub-leading singularities

• Augmented Recursion need formalism to work a off-shell (partially) but still use helicity information: -light -cone gauge methods -carry out a BCFW shift

• • • relies upon working off-shell , (a little as possible) uses off-shell currents from Yang-Mills Berends-Giele, Kosower, Mahlon assumes KLT , close to off-shell produces very cumbersome but, usable, result Alston, Dunbar and Perkins please, please trivialise http: //pyweb. swan. ac. uk/~dunbar/graviton. html

Soft Theorems? ? ? • • • all-plus satisfies theorem single minus satisfies theorem when negative leg single minus fails sub-leading result Bern, Davies, Nohle He, Huang, and Wen

Soft-Limit is a coupled BCFW shift t • sub-leading directly related to double poles

Conclusions • • soft theorems seem good at sub-leading fail at sub-leading constraints equivalent to collinear non-supersymmetric a long way from maximally

N=4